Navier-Stokes blow-up rates in certain Besov spaces whose regularity exceeds the critical value by $\boldsymbol{\epsilon \in [1,2]}$

Abstract

For a solution $u$ to the Navier-Stokes equations in spatial dimension $n\geq3$ which blows up at a finite time $T>0$, we prove the blowup estimate ${\|u(t)\|}_{\dot{B}_{p,q}^{s_{p}+\epsilon}(\mathbb{R}^n)}\gtrsim_{\varphi,\epsilon,(p\vee q\vee 2)}{(T-t)}^{-\epsilon/2}$ for all $\epsilon\in[1,2)$ and $p,q\in[1,\frac{n}{2-\epsilon})$, where $s_{p}:=-1+\frac{n}{p}$ is the scaling-critical regularity, and $\varphi$ is the cutoff function used to define the Littlewood-Paley projections. For $\epsilon =2$, we prove the same type of estimate but only for $q=1$: ${\|u(t)\|}_{\dot{B}_{p,1}^{s_{p}+2}(\mathbb{R}^n)}\gtrsim_{\varphi,(p\vee 2)}{(T-t)}^{-1}$ for all $p\in [1,\infty)$. Under the additional restriction that $p,q\in[1,2]$ and $n=3$, these blowup estimates are implied by those first proved by Robinson, Sadowski and Silva (J. Math. Phys., 2012) for $p=q=2$ in the case $\epsilon\in(1,2)$, and by McCormick, Olson, Robinson, Rodrigo, Vidal-L\'{o}pez and Zhou (SIAM J. Math. Anal., 2016) for $p=2$ in the cases $(\epsilon,q)=(1,2)$ and $(\epsilon,q)=(2,1)$.